2014/07/17 by Ayşe Hümeyra Bilge, Bilge, Ayse Humeyra, Yunus Özdemir +1 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1407.4839
openalex publication_date 2014/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let y(t) be a smooth sigmoidal curve, y(n)(t) be its nth derivative, \tm,i\ and \ta,i\, i=1,2,… be the set of points where respectively the derivatives of odd and even order reach their extreme values. The "critical point of the sigmoidal curve" is defined to be the common limit of the sequences \tm,i\ and \ta,i\, provided that the limit exists. We prove that if f(t)=(dy)/(dt) is an even function such that the magnitude of the analytic representation |fA(t)|=|f(t)+ifh(t)|, where fh(t) is the Hilbert transform of f(t), is monotone on (0,∞), then the point t=0 is the critical point in the sense above. For the general case, where f(t)=(dy)/(dx) is not even, we prove that if |fA(t)| monotone on (0,∞) and if the phase of its Fourier transform F(ω) has a limit as ω→ ±∞, then t=0 is still the critical point but as opposed to the previous case, the maximum of f(t) is located away from t=0. We compute the Fourier transform of the generalized logistic growth functions and illustrate the notions above on these examples.